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Lesson by lesson podcasts for teachers of Illustrative Mathematics®.
(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)
Alright. So are you ready to, like, ditch the shovels and, like, grab your graphing calculators? Because today, we are going to be diving into the world of landscaping, but, like, landscaping with inequalities.
Speaker 2:That's right. We're taking illustrative math lesson 22, which, you know, focuses on linear inequalities, and we're showing you how to help your students to apply these to, like, real world situations.
Speaker 1:Yeah. I'm already intrigued. We're not just talking about abstract equations here. Right? This is about equipping our students to be, like, landscaping consultants making budgeting decisions and, like, navigating all these design dilemmas all through the power of math.
Speaker 2:And, you know, what's really fascinating about this lesson is how it takes students from that world of linear equations Mhmm. You know, representing those nice, neat, fixed costs and then throws them into this, like, exciting, messy world of inequalities where budgets exist and profits, you know, need maximizing.
Speaker 1:Okay. So we're talking about, like, real world constraints. Right? Not just finding an answer, but, like, understanding the range of possibilities.
Speaker 2:Exactly.
Speaker 1:The lesson plan mentions a homeowner grappling with some classic landscaping choices. You know, sod versus flower beds, artificial turf versus gravel. I can already see the dollar signs flashing before my eyes.
Speaker 2:Exactly. And that's where the graphing comes in. Right? We're not just plotting lines for the fun of it here. Your students are gonna be creating visual representations of budget scenarios, those all important boundary lines, and get this, they'll be shading in entire regions on the graph to see all the viable options.
Speaker 1:So it's like coloring in the possibilities, but with a purpose.
Speaker 2:With a purpose. Exactly.
Speaker 1:Now this is why I'm curious. If all those shaded points are potential solutions, how do we guide our students to, like, pick the best one given the constraints? I mean, they can't just pick any point in the shaded region. Right?
Speaker 2:Absolutely not. And the beauty of this lesson is that it doesn't stop at simply, you know, identifying solutions. It pushes students to think critically about what those solutions actually mean in the real world.
Speaker 1:So it's not enough to just, like, stay under budget. Right? You have to think practically, like, would a homeowner really want, like, a tiny patch of turf surrounded by a sea of gravel even if it, like, technically fits the budget? I don't think so.
Speaker 2:Exactly. And the lesson emphasizes this point beautifully. It reminds us that a mathematically sound solution, right, might not always be the most, practical or, like, aesthetically pleasing choice.
Speaker 1:It's about finding that sweet spot where math meets real world considerations.
Speaker 2:Now shifting gears for a moment, let's talk about the second scenario that's presented in this lesson. Remember that vendor at the Saturday market trying to figure out how many necklaces and bracelets to make to hit their profit target? That's where this deep dive, I think, takes a really interesting turn. I've actually seen this kind of problem, like, stump some pretty experienced math teachers.
Speaker 1:Oh, I'm hooked. So we're not just talking about landscaping anymore. We're talking about equipping our students with math skills that can be applied to, like entrepreneurship. I love it.
Speaker 2:Precisely. And this scenario really highlights how versatile inequalities can be. Instead of square footage and landscaping materials, we're now dealing with units of necklaces, bracelets, and, like, different profit margins.
Speaker 1:Yeah. This feels like a whole new level of problem solving.
Speaker 2:It is. And this is where that moment, I think, often hits students. They realize that those shaded regions on their graphs aren't just like abstract mathematical concepts. They represent a menu of viable options for their entrepreneurial venture.
Speaker 1:Okay. I can see how that would be, like, empowering for students to see that they can use math to not only model but also, like, strategize for success.
Speaker 2:And it leads to some really engaging classroom discussions. For instance, should the vendor focus on selling more necklaces, which have a higher profit margin per unit, or aim to sell more bracelets, which might be an easier sell at a lower price point?
Speaker 1:Oh, I can imagine the debates that would spark. There's no one right answer. So students have to, like, justify their reasoning and weigh different factors. It's real world decision making in action.
Speaker 2:Exactly. Now this is where those potential misconceptions we kind of touched on earlier become even more critical to address. Because if students misinterpret what the graph represents or, like, make faulty assumptions about the solution set, it can derail their entire decision making process.
Speaker 1:Yeah. It's like giving them the keys to a powerful car, but making sure they know how drive it before letting them loose on the highway. We want them to use these tools effectively.
Speaker 2:Precisely. Speaking of tools, there's an interesting note in the lesson plan about not using graphing technology for this particular lesson. I'm curious to get your take on that.
Speaker 1:You know, I think there's real value in that, like, hands on approach, especially when it comes to building a solid understanding of inequalities. When students have to graph by hand, plotting those points, figuring out intercepts, they're forced to really engage with the process.
Speaker 2:I agree. It prevents them from becoming overly reliant on technology and helps them develop a deeper, more intuitive grasp of what those lines and shaded regions truly represent. It's about empowering them to reason through problems, not just punch numbers into a calculator and hope for the best.
Speaker 1:And let's be honest. There's a certain satisfaction that comes from, like, finding solutions through your own logic and calculations. It builds confidence and solidifies those, like, foundational math skills, but the outline doesn't stop at landscaping or market stalls. It encourages us to think beyond these scenarios. So where else in the real world could our students apply these inequality solving superpowers?
Speaker 2:Oh, the possibilities are practically endless.
Speaker 1:I love that phrase, inequality solving superpowers. What other, like, real world villains can we tackle with these?
Speaker 2:Well, think about something like, planning a fundraising event for your class. You got a budget. You got a list of expenses that's as long as your arm and a fundraising goal you need to
Speaker 1:hit. Right.
Speaker 2:Inequalities are the perfect tool to model all of that.
Speaker 1:Oh, totally. Your your students could explore different ticket processes, the impact of donation levels, all while making sure they, like, stay in the black. It's like they become financial strategies.
Speaker 2:I've even seen teachers use inequalities to explore resource allocation in video games.
Speaker 1:What? Wait. That's amazing. It's those tough choices we all make in games, but now I see how it ties back to math.
Speaker 2:It's about understanding trade offs and making strategic decisions within those limitations.
Speaker 1:You know what's striking me is how this lesson takes something that could feel very abstract, these symbols and shaded regions on a graph, and transforms them into tools for understanding real world situations.
Speaker 2:Exactly. It's not just about finding the right answer on a worksheet. It's about equipping students to analyze, to strategize, and to make informed decisions when faced with constraints. And those are skills that go way beyond the math classroom.
Speaker 1:A 100%. Huge thank you for taking this deep dive with me. This episode has been, really eye opening.
Speaker 2:My pleasure.
Speaker 1:And to all our listeners out there, we hope this deep dive into illustrative math lesson 22 has sparked some new ideas for your classrooms. Remember, inequalities aren't just about math. They're about empowering your students to navigate a world full of choices, constraints, and, endless possibilities. Until next time.